Palace of Catalan Music (S. Adams/GETTY)

## Geometry, Physics, and Representation Theory Seminar

### RTG Group Seminar

#### Fall 2019, Northeastern

 Meeting weekly on Thursdays 2:50-3:50pm in 509/511 Lake Hall at Northeastern. If you are not at Northeastern, but would like to recieve announcements, join the mailing list. If you have a question or would like to speak at seminar, email

### Schedule

 Date Speaker Title Sep 5 Laure Flapan (MIT) Complete families of indecomposable non-simple abelian varieties An abelian variety $A$ is indecomposable but not simple if $A$ is isogenous to a product of smaller-dimensional abelian varieties but is not itself a product. We explore how to construct complete families of abelian varieties with this property and discuss implications for the question of understanding the possible fundamental groups of fibered projective varieties. Sep 12 Daoji Huang (Cornell) Bruhat Atlas on Stratified Manifolds in Coordinates A stratified smooth variety $M$ admits a Bruhat atlas if it can be covered by open charts isomorphic to opposite Bruhat cells in some Kac-Moody flag manifold via stratified isomorphisms. In this talk, I will present two cases where the Kac-Moody is finite or affine type, which allows explicit computation in coordinates using Bott-Samelson maps and other familiar techniques in Schubert calculus. Sep 19 Chris Gerig (Harvard) Probing 4-manifolds with near-symplectic forms Most closed 4-manifolds do not admit symplectic forms, but most admit "near-symplectic forms", certain closed 2-forms which are symplectic outside of a collection of circles. This provides a gateway from the symplectic world to the non-symplectic world. Just like the Seiberg-Witten (SW) invariants, there are invariants in terms of J-holomorphic curves that are compatible with the near-symplectic form. Although the SW invariants don't apply to (potentially exotic) 4-spheres, nor do these spheres admit near-symplectic forms, there is still a way to bring in near-symplectic techniques. Sep 26 Matt Hogancamp (Northeastern) Trace of the Hecke category Recall that the cocenter (or trace, or $HH_0$) of an algebra $A$ is the algebra modulo commutators. It is well known that the direct sum of cocenters of symmetric group algebras $Q[S_n]$ forms an algebra isomorphic to the ring of symmetric functions in infinitely many variables. There is a $q$-deformation of this fact: the direct sum of cocenters of type $A$ Hecke algebras forms an algebra isomorphic to the ring of symmetric functions with an additional formal parameter $q$. In this talk I will discuss a categorification of this fact, in which the Hecke algebra gets replaced by the category of Soergel bimodules (or Hecke category"). I will present an explicit dg model for the categorical cocenter. Miraculously, the cocenters of Hecke categories can be calculated (in type $A$, anyway) as the derived categories of explicit wreath product algebras. Finally, I plan to sketch how this gives rise to a well-behaved notion of Khovanov-Rozansky link homology for links in a solid torus (which was the primary motivation for this work). This is joint with Eugene Gorsky and Paul Wedrich. Oct 3 Noah Giansiracusa (Bentley) A matroidal view of group representations Tropical geometry provides a convenient algebraic framework for matroids, and in this talk I'll present recent work with a student, Jacob Manaker, where we use this language to explore representations of groups on tropical linear spaces, rather than on vector spaces over a field. This brings up some intriguing combinatorial aspects of classical representation theory, though we are only at the first steps of this story and hope to interest others in joining this project. Oct 10 Peter Koroteev (UC Berkeley) $q$-Opers, $QQ$-systems, and Bethe ansatz A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers - connections on the projective line with extra structure. I shall describe a deformation of this correspondence for $\operatorname{SL}(N)$. I will introduce a difference equation version of opers called $q$-opers and prove a $q$-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted $q$-opers with regular singularities on the projective line. The so-called quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars-Schneider model may be viewed as a special case of the $q$-Langlands correspondence. Some applications of $q$-opers to the equivariant quantum K-theory of the cotangent bundles to partial flag varieties will be discussed as well as generalizations of our constructions to an arbitrary simply connected complex simple Lie group $G$. Oct 17 RTG member presentations A succession of short presentations on the research conducted by RTG group members. Oct 24 RTG member presentations A succession of short presentations on the research conducted by RTG group members. Oct 31 No Seminar. Nov 7 Philsang Yoo (Yale) Integrable Hierarchy From the BV Formalism The KdV equation is a nonlinear partial differential equation describing waves on shallow water surfaces. In spite of its nonlinearity, this is exactly solvable as it admits a surprisingly rich structure, that is, it is a part of the so-called KdV hierarchy. From a completely different direction, for a smooth projective variety $X$, one can consider a generating function of Gromov--Witten invariants of $X$. Witten conjectured and Kontsevich first proved the mysterious claim that when $X$ is a point, the generating function is governed by the KdV hierarchy. I will explain a program toward understanding what happens when $X$ is general. This talk is on the first step for dispersionless integrable hierarchy, which is a joint work with Weiqiang He, Si Li, and Xinxing Tang. Nov 14 Susan Tolman (UIUC) Toric degeneration and symplectic rigidity This talk is based on joint work with Milena Pabiniak. We say that a family of symplectic manifolds satisfies symplectic rigidity if they are classified up to symplectomorphism by their cohomology ring and the cohomology class of the symplectic form. We use toric degeneration to construct new (non-equivariant) symplectomorphisms between certain smooth toric manifolds. This enables us to show that symplectic rigidity holds for a large family of Bott manifolds. In particular, it holds for the family of symplectic toric manifolds whose integral cohomology is isomorphic to that of the product of n two-spheres. Nov 21 Si Li (Tsinghua) Localized Effective Theory, Semi-classical Approximation and Algebraic Index We describe a general framework to study the quantum geometry of $\sigma$-models when they are effectively localized to small quantum fluctuations around constant maps. Such effective theories have exact descriptions at all loops in terms of target geometry and can be rigorously formulated. We illustrate this idea by the example of topological quantum mechanics which will lead to an explicit construction of the universal trace map on periodic cyclic chains of matrix Weyl algebras. As an application, we explain how to implement the idea of exact semi-classical approximation into a proof of the algebraic index theorem using Getzler’s Gauss-Manin connection. This is joint work with Zhengping Gui and Kai Xu. Nov 28 No Seminar. Thanksgiving Break. Dec 5 Reyer Sjamaar (Cornell) Stacky Hamiltonian actions and symplectic reduction We introduce the notion of a Hamiltonian action of an étale Lie group stack on an étale symplectic stack and establish versions of some basic theorems of symplectic geometry in this context, such as the Kirwan convexity theorem and the Mayer-Marsden-Weinstein symplectic reduction theorem. This is joint work with Benjamin Hoffman. End of Semester (Seminar Resumes Spring 2020)

#### Previous Semesters

 Spring 2019 Fall 2018 Spring 2018 Fall 2017 Spring 2017 Fall 2016 Spring 2016 Fall 2015 Spring 2015